def sols_for_one_pow(n):
"""
Counts all solutions to the diophantine equation 1/a+1/b=p/10^n
All variables are positive integers, n is fixed
"""
tot = 0
pow5 = [5**i for i in xrange(n+2)]
pow2 = [2**i for i in xrange(n+2)]
for e1 in xrange(n+1):
for e2 in xrange(n+1):
a,b = pow5[e1],pow2[e2]
mul = pow5[n-e1]*pow2[n-e2]
tot += count_divs(shanks_factorize(mul*(a*b+1)))
if e1 and e2:
tot += count_divs(shanks_factorize(mul*(a+b)))
return tot
def m_search(n,k,m=1,phi=1,lowerInd=0):
global primes,mask,pcap
s = 0
if k == 0:
M = m*phi-phi+m
bnd = int(M**.25)
pi = bisect_right(primes,n/m)
if n/m > pcap:
f = shanks_factorize(M,primes[:15])
for d in divisors([[bas,f[bas]] for bas in f]):
q,r = divmod(d-phi,m-phi)
if r == 0 and q > primes[lowerInd] and q*m <= n and is_prime(q,pcap-1,mask):
if (q*m-1) % (q*m - (q-1)*phi) == 0:
s += q*m
else:
for q in primes[lowerInd+1:pi+1]:
if q*m <= n and (q*m-1) % (q*m - (q-1)*phi) == 0:
s+= q*m
# print q*m,q,m
return s
bound = int((n/m)**(1./(k+1)))
i=lowerInd+1
for i,p in enumerate(primes[lowerInd+1:],lowerInd+1):
if p > bound:
break
else:
s += m_search(n,k-1,m*p,phi*(p-1),i)
return s
def cheat(n):
nums = np.loadtxt("e245.txt",dtype=np.int64)
nums = nums[:,1]
for i in xrange(nums.shape[0]):
f = shanks_factorize(nums[i])
if len(f) >= 4:
print nums[i],f
return sum(nums[np.where(nums<=n)])
def solve_1(cap,mod=10**9+7):
primes = primes_and_mask(cap)[0]
R = [1 for i in xrange(cap+1)]
N = [i**4+4 for i in xrange(cap+1)]
sq = [0,0,0,0]
for p in primes:
if p == 2:
for i in xrange(1,cap+1):
mul = 1
while N[i]%2==0:
N[i]/=2
mul*=2
if mul>1:
R[i] = (R[i]*(mul+1))%mod
continue
sq[0] = mod_sqrt(p-4,p)
if not sq[0]:
continue
sq[2] = mod_sqrt(sq[0],p)
sq[3] = mod_sqrt(p-sq[0],p)
sq[0] = (p-sq[3])%p
sq[1] = (p-sq[2])%p
# print p,sq
for q in xrange(0,cap+1,p):
for r in sq:
if r and r <= cap-q:
mul=1
while N[q+r]%p==0:
mul *= p
N[q+r]/=p
if mul > 1:
R[q+r] = (R[q+r]*(mul+1))%mod
print "Completed seiving."
c=0
s = 0
for i,r in enumerate(R):
if i==0:
continue
n = N[i]
if n ==1:
s = (s + r -i**4-4)%mod
continue
factors = shanks_factorize(n)
for f in factors:
if f>1:
r = (r*(f**factors[f]+1))%mod
if len(factors)>=2:
c +=1
s = (s + r -i**4-4)%mod
print s
print c
return s
def find_pairs(n):
global primes,mask
s = 0
pairs = {}
for p in primes[1:]:
num = p**2-p+1
f = shanks_factorize(num,primes[:15])
for k in divisors([[bas,f[bas]] for bas in f] ):
q = num/k - p + 1
if q > 0 and q > p and q*p <= n and is_prime(q,pcap-1,mask):
s += p*q
# print p,q
if p not in pairs:
pairs[p] = [q]
else:
pairs[p].append(q)
print "Sum over semi-primes %d" % s
#now do backtracking pair-combos
for p1 in pairs:
for p2 in pairs[p1]:
s += cores_backtrack(n/p1/p2,pairs,[p1,p2])
return s
def r_slow(n): r=1 factors = shanks_factorize(n) for f in factors: r *= f**factors[f]+1 return r-n
#k choose i, then i choose j+i-k
target[k-1] += C(k,i) * C(i, j+i-k) * a[i-1]*b[j-1]
return target
for i in xrange(200):
p_rows.append(pascal_row(i+2))
bound = 10**16
classes, primes = prime_equivalences(bound)
tot = 0
for c in classes:
g_rep = p_rows[c[0]-1]
for el in c[1:]:
if not el:
break
else:
g_rep = combine_gozinta(g_rep, p_rows[el-1])
g = sum(g_rep)
if g > bound:
continue
# print g, c
fact = shanks_factorize(g)
rep = fact.values()
trunc_rep = sorted(rep)[::-1]
for i in xrange(len(rep), len(c)):
trunc_rep.append(0)
if trunc_rep == c:
print g, fact, trunc_rep, c
tot += g
print "Total "+str(tot)
def f(n): f = shanks_factorize(n) return max(f.keys())